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Full text of "Analytical Mechanics"

MOTION OF A PARTICLE                       289
4-Vi/v^                                         d^uf    dv     dv
tnen                          — = —- = -—v
dd2     dd    dur '
and                            v^L + u' = 0
du
Separating the variable and integrating
i^/ = ^v____ Integrating again
ni-
cos"1 —- = e + 8
A
or           uf = A cos
Let         u' = A when 0 =0,    then   5 = 0.    Therefore
u' = A cos 0                                 (3)
is a solution of equation (2).   Substituting the value of uf in equation (3),
u=f- + Acos6                         (4)
Ai"
and replacing u by its value
r=-----^—                               (5)
1 — e cos 6
h2A            1
where                         e =-------, p = — — -     ,            •    (6)
At                     A
Equation (5) is the well-known equation of a conic section. Therefore the orbit is a conic section with an eccentricity
i *       h*A equal to------•
M The expression for the velocity at any point" of the orbit
C!T        dd may be obtained by substituting the values of—and —. which